Data Analysis Siegmund Brandt Statistical and Computational Methods for Scientists and Engineers Fourth Edition Data Analysis Siegmund Brandt Data Analysis Statistical and Computational Methods for Sc.
Siegmund Brandt Data Analysis Statistical and Computational Methods for Scientists and Engineers Fourth Edition Data Analysis Siegmund Brandt Data Analysis Statistical and Computational Methods for Scientists and Engineers Fourth Edition Translated by Glen Cowan 123 Siegmund Brandt Department of Physics University of Siegen Siegen, Germany Additional material to this book can be downloaded from http://extras.springer.com ISBN 978-3-319-03761-5 ISBN 978-3-319-03762-2 (eBook) DOI 10.1007/978-3-319-03762-2 Springer Cham Heidelberg New York Dordrecht London Library of Congress Control Number: 2013957143 © Springer International Publishing Switzerland 2014 This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer Permissions for use may be obtained through RightsLink at the Copyright Clearance Center Violations are liable to prosecution under the respective Copyright Law The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made The publisher makes no warranty, express or implied, with respect to the material contained herein Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) Preface to the Fourth English Edition For the present edition, the book has undergone two major changes: Its appearance was tightened significantly and the programs are now written in the modern programming language Java Tightening was possible without giving up essential contents by expedient use of the Internet Since practically all users can connect to the net, it is no longer necessary to reproduce program listings in the printed text In this way, the physical size of the book was reduced considerably The Java language offers a number of advantages over the older programming languages used in earlier editions It is object-oriented and hence also more readable It includes access to libraries of user-friendly auxiliary routines, allowing for instance the easy creation of windows for input, output, or graphics For most popular computers, Java is either preinstalled or can be downloaded from the Internet free of charge (See Sect 1.3 for details.) Since by now Java is often taught at school, many students are already somewhat familiar with the language Our Java programs for data analysis and for the production of graphics, including many example programs and solutions to programming problems, can be downloaded from the page extras.springer.com I am grateful to Dr Tilo Stroh for numerous stimulating discussions and technical help The graphics programs are based on previous common work Siegen, Germany Siegmund Brandt v Contents Preface to the Fourth English Edition v List of Examples xv Frequently Used Symbols and Notation xix Introduction 1.1 Typical Problems of Data Analysis 1.2 On the Structure of this Book 1.3 About the Computer Programs Probabilities 2.1 Experiments, Events, Sample Space 2.2 The Concept of Probability 2.3 Rules of Probability Calculus: Conditional Probability 2.4 Examples 2.4.1 Probability for n Dots in the Throwing of Two Dice 2.4.2 Lottery Out of 49 2.4.3 Three-Door Game 1 10 11 11 12 13 Random Variables: Distributions 3.1 Random Variables 3.2 Distributions of a Single Random Variable 3.3 Functions of a Single Random Variable, Expectation Value, Variance, Moments 3.4 Distribution Function and Probability Density of Two Variables: Conditional Probability 3.5 Expectation Values, Variance, Covariance, and Correlation 15 15 15 17 25 27 vii viii Contents 3.6 3.7 3.8 More than Two Variables: Vector and Matrix Notation 30 Transformation of Variables 33 Linear and Orthogonal Transformations: Error Propagation 36 Computer Generated Random Numbers: The Monte Carlo Method 4.1 Random Numbers 4.2 Representation of Numbers in a Computer 4.3 Linear Congruential Generators 4.4 Multiplicative Linear Congruential Generators 4.5 Quality of an MLCG: Spectral Test 4.6 Implementation and Portability of an MLCG 4.7 Combination of Several MLCGs 4.8 Generation of Arbitrarily Distributed Random Numbers 4.8.1 Generation by Transformation of the Uniform Distribution 4.8.2 Generation with the von Neumann Acceptance–Rejection Technique 4.9 Generation of Normally Distributed Random Numbers 4.10 Generation of Random Numbers According to a Multivariate Normal Distribution 4.11 The Monte Carlo Method for Integration 4.12 The Monte Carlo Method for Simulation 4.13 Java Classes and Example Programs Some Important Distributions and Theorems 5.1 The Binomial and Multinomial Distributions 5.2 Frequency: The Law of Large Numbers 5.3 The Hypergeometric Distribution 5.4 The Poisson Distribution 5.5 The Characteristic Function of a Distribution 5.6 The Standard Normal Distribution 5.7 The Normal or Gaussian Distribution 5.8 Quantitative Properties of the Normal Distribution 5.9 The Central Limit Theorem 5.10 The Multivariate Normal Distribution 5.11 Convolutions of Distributions 5.11.1 Folding Integrals 5.11.2 Convolutions with the Normal Distribution 5.12 Example Programs 41 41 42 44 45 47 50 52 55 55 58 62 63 64 66 67 69 69 72 74 78 81 84 86 88 90 94 100 100 103 106 Contents ix Samples 6.1 Random Samples Distribution of a Sample Estimators 6.2 Samples from Continuous Populations: Mean and Variance of a Sample 6.3 Graphical Representation of Samples: Histograms and Scatter Plots 6.4 Samples from Partitioned Populations 6.5 Samples Without Replacement from Finite Discrete Populations Mean Square Deviation Degrees of Freedom 6.6 Samples from Gaussian Distributions: χ -Distribution 6.7 χ and Empirical Variance 6.8 Sampling by Counting: Small Samples 6.9 Small Samples with Background 6.10 Determining a Ratio of Small Numbers of Events 6.11 Ratio of Small Numbers of Events with Background 6.12 Java Classes and Example Programs The Method of Maximum Likelihood 7.1 Likelihood Ratio: Likelihood Function 7.2 The Method of Maximum Likelihood 7.3 Information Inequality Minimum Variance Estimators Sufficient Estimators 7.4 Asymptotic Properties of the Likelihood Function and Maximum-Likelihood Estimators 7.5 Simultaneous Estimation of Several Parameters: Confidence Intervals 7.6 Example Programs 109 109 111 115 122 127 130 135 136 142 144 147 149 153 153 155 157 164 167 173 Testing Statistical Hypotheses 8.1 Introduction 8.2 F -Test on Equality of Variances 8.3 Student’s Test: Comparison of Means 8.4 Concepts of the General Theory of Tests 8.5 The Neyman–Pearson Lemma and Applications 8.6 The Likelihood-Ratio Method 8.7 The χ -Test for Goodness-of-Fit 8.7.1 χ -Test with Maximal Number of Degrees of Freedom 8.7.2 χ -Test with Reduced Number of Degrees of Freedom 8.7.3 χ -Test and Empirical Frequency Distribution 175 175 177 180 185 191 194 199 199 200 200 I Statistical Tables 509 Table I.5: Quantiles xP = Ω (P ) of the normal distribution P =√ xP 2π −xP exp(−x /2) dx P 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.000 0.126 0.253 0.385 0.524 0.674 0.842 1.036 1.282 1.645 0.013 0.138 0.266 0.399 0.539 0.690 0.860 1.058 1.311 1.695 0.025 0.151 0.279 0.412 0.553 0.706 0.878 1.080 1.341 1.751 0.038 0.164 0.292 0.426 0.568 0.722 0.896 1.103 1.372 1.812 0.050 0.176 0.305 0.440 0.583 0.739 0.915 1.126 1.405 1.881 0.063 0.189 0.319 0.454 0.598 0.755 0.935 1.150 1.440 1.960 0.075 0.202 0.332 0.468 0.613 0.772 0.954 1.175 1.476 2.054 0.088 0.215 0.345 0.482 0.628 0.789 0.974 1.200 1.514 2.170 0.100 0.228 0.358 0.496 0.643 0.806 0.994 1.227 1.555 2.326 0.113 0.240 0.372 0.510 0.659 0.824 1.015 1.254 1.598 2.576 P 0.90 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.645 1.695 1.751 1.812 1.881 1.960 2.054 2.170 2.326 2.576 1.650 1.701 1.757 1.818 1.888 1.969 2.064 2.183 2.346 2.612 1.655 1.706 1.762 1.825 1.896 1.977 2.075 2.197 2.366 2.652 1.660 1.711 1.768 1.832 1.903 1.986 2.086 2.212 2.387 2.697 1.665 1.717 1.774 1.838 1.911 1.995 2.097 2.226 2.409 2.748 1.670 1.722 1.780 1.845 1.919 2.005 2.108 2.241 2.432 2.807 1.675 1.728 1.787 1.852 1.927 2.014 2.120 2.257 2.457 2.878 1.680 1.734 1.793 1.859 1.935 2.024 2.132 2.273 2.484 2.968 1.685 1.739 1.799 1.866 1.943 2.034 2.144 2.290 2.512 3.090 1.690 1.745 1.805 1.873 1.951 2.044 2.157 2.308 2.543 3.291 P 0.990 0.991 0.992 0.993 0.994 0.995 0.996 0.997 0.998 0.999 2.576 2.612 2.652 2.697 2.748 2.807 2.878 2.968 3.090 3.291 2.579 2.616 2.656 2.702 2.753 2.814 2.886 2.978 3.105 3.320 2.583 2.620 2.661 2.706 2.759 2.820 2.894 2.989 3.121 3.353 2.586 2.624 2.665 2.711 2.765 2.827 2.903 3.000 3.138 3.390 2.590 2.628 2.669 2.716 2.770 2.834 2.911 3.011 3.156 3.432 2.594 2.632 2.674 2.721 2.776 2.841 2.920 3.023 3.175 3.481 2.597 2.636 2.678 2.727 2.782 2.848 2.929 3.036 3.195 3.540 2.601 2.640 2.683 2.732 2.788 2.855 2.938 3.048 3.216 3.615 2.605 2.644 2.687 2.737 2.794 2.863 2.948 3.062 3.239 3.719 2.608 2.648 2.692 2.742 2.801 2.870 2.958 3.076 3.264 3.891 510 I Statistical Tables Table I.6: χ -distribution F (χ ) χ2 F (χ ) = f (χ ; f ) dχ f χ2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 20.0 0.248 0.345 0.416 0.473 0.520 0.561 0.597 0.629 0.657 0.683 0.843 0.917 0.954 0.975 0.986 0.992 0.995 0.997 0.998 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.049 0.095 0.139 0.181 0.221 0.259 0.295 0.330 0.362 0.393 0.632 0.777 0.865 0.918 0.950 0.970 0.982 0.989 0.993 0.996 0.998 0.998 0.999 0.999 1.000 1.000 1.000 1.000 1.000 0.008 0.022 0.040 0.060 0.081 0.104 0.127 0.151 0.175 0.199 0.428 0.608 0.739 0.828 0.888 0.928 0.954 0.971 0.981 0.988 0.993 0.995 0.997 0.998 0.999 0.999 1.000 1.000 1.000 0.001 0.005 0.010 0.018 0.026 0.037 0.049 0.062 0.075 0.090 0.264 0.442 0.594 0.713 0.801 0.864 0.908 0.939 0.960 0.973 0.983 0.989 0.993 0.995 0.997 0.998 0.999 0.999 1.000 0.000 0.001 0.002 0.005 0.008 0.012 0.017 0.023 0.030 0.037 0.151 0.300 0.451 0.584 0.694 0.779 0.844 0.891 0.925 0.949 0.965 0.977 0.984 0.990 0.993 0.996 0.997 0.998 0.999 0.000 0.000 0.001 0.001 0.002 0.004 0.006 0.008 0.011 0.014 0.080 0.191 0.323 0.456 0.577 0.679 0.762 0.826 0.875 0.912 0.938 0.957 0.970 0.980 0.986 0.991 0.994 0.996 0.997 0.000 0.000 0.000 0.000 0.001 0.001 0.002 0.003 0.004 0.005 0.040 0.115 0.220 0.340 0.460 0.571 0.667 0.747 0.811 0.861 0.899 0.928 0.949 0.964 0.975 0.983 0.988 0.992 0.994 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.001 0.002 0.019 0.066 0.143 0.242 0.353 0.463 0.567 0.658 0.735 0.798 0.849 0.888 0.918 0.941 0.958 0.970 0.979 0.985 0.990 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.009 0.036 0.089 0.166 0.260 0.363 0.466 0.563 0.650 0.724 0.787 0.837 0.878 0.909 0.933 0.951 0.965 0.975 0.982 10 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.004 0.019 0.053 0.109 0.185 0.275 0.371 0.468 0.560 0.642 0.715 0.776 0.827 0.868 0.900 0.926 0.945 0.960 0.971 I Statistical Tables 511 Table I.7: Quantiles χP2 of the χ -distribution χP2 P= f (χ ; f ) dχ f 0.900 0.950 2.706 3.841 4.605 5.991 6.251 7.815 7.779 9.488 9.236 11.070 10.645 12.592 12.017 14.067 13.362 15.507 14.684 16.919 10 15.987 18.307 11 17.275 19.675 12 18.549 21.026 13 19.812 22.362 14 21.064 23.685 15 22.307 24.996 16 23.542 26.296 17 24.769 27.587 18 25.989 28.869 19 27.204 30.144 20 28.412 31.410 30 40.256 43.773 40 51.805 55.758 50 63.167 67.505 60 74.397 79.082 70 85.527 90.531 80 80.000 101.879 90 107.565 113.145 100 118.498 124.342 P 0.990 6.635 9.210 11.345 13.277 15.086 16.812 18.475 20.090 21.666 23.209 24.725 26.217 27.688 29.141 30.578 32.000 33.409 34.805 36.191 37.566 50.892 63.691 76.154 88.379 100.425 112.329 124.116 135.807 0.995 7.879 10.597 12.838 14.860 16.750 18.548 20.278 21.955 23.589 25.188 26.757 28.300 29.819 31.319 32.801 34.267 35.718 37.156 38.582 39.997 53.672 66.766 79.490 91.952 104.215 116.321 128.299 140.169 0.999 10.828 13.816 16.266 18.467 20.515 22.458 24.322 26.124 27.877 29.588 31.264 32.909 34.528 36.123 37.697 39.252 40.790 42.312 43.820 45.315 59.703 73.402 86.661 99.607 112.317 124.839 137.208 149.449 512 I Statistical Tables Table I.8: Quantiles FP of the F -distribution 0.900 = P FP = f (F ; f1 , f2 ) dF f2 10 39.86 8.526 5.538 4.545 4.060 3.776 3.589 3.458 3.360 3.285 49.50 9.000 5.462 4.325 3.780 3.463 3.257 3.113 3.006 2.924 53.59 9.162 5.391 4.191 3.619 3.289 3.074 2.924 2.813 2.728 55.83 9.243 5.343 4.107 3.520 3.181 2.961 2.806 2.693 2.605 f1 57.24 58.20 9.293 9.326 5.309 5.285 4.051 4.010 3.453 3.405 3.108 3.055 2.883 2.827 2.726 2.668 2.611 2.551 2.522 2.461 FP 0.950 = P = 58.91 9.349 5.266 3.979 3.368 3.014 2.785 2.624 2.505 2.414 59.44 9.367 5.252 3.955 3.339 2.983 2.752 2.589 2.469 2.377 59.86 9.381 5.240 3.936 3.316 2.958 2.725 2.561 2.440 2.347 10 60.19 9.392 5.230 3.920 3.297 2.937 2.703 2.538 2.416 2.323 240.5 19.38 8.812 5.999 4.772 4.099 3.677 3.388 3.179 3.020 10 241.9 19.40 8.786 5.964 4.735 4.060 3.637 3.347 3.137 2.978 f (F ; f1 , f2 ) dF f2 10 161.4 18.51 10.13 7.709 6.608 5.987 5.591 5.318 5.117 4.965 199.5 19.00 9.552 6.944 5.786 5.143 4.737 4.459 4.256 4.103 215.7 19.16 9.277 6.591 5.409 4.757 4.347 4.066 3.863 3.708 224.6 19.25 9.117 6.388 5.192 4.534 4.120 3.838 3.633 3.478 f1 230.2 234.0 19.30 19.33 9.013 8.941 6.256 6.163 5.050 4.950 4.387 4.284 3.972 3.866 3.687 3.581 3.482 3.374 3.326 3.217 236.8 19.35 8.887 6.094 4.876 4.207 3.787 3.500 3.293 3.135 238.9 19.37 8.845 6.041 4.818 4.147 3.726 3.438 3.230 3.072 I Statistical Tables 513 Table I.8: (continued) 0.975 = P FP = f (F ; f1 , f2 ) dF f2 10 647.8 38.51 17.44 12.22 10.01 8.813 8.073 7.571 7.209 6.937 799.5 39.00 16.04 10.65 8.434 7.260 6.542 6.059 5.715 5.456 864.2 39.17 15.44 9.979 7.764 6.599 5.890 5.416 5.078 4.826 899.6 39.25 15.10 9.605 7.388 6.227 5.523 5.053 4.718 4.468 f1 921.8 937.1 39.30 39.33 14.88 14.73 9.364 9.197 7.146 6.978 5.988 5.820 5.285 5.119 4.817 4.652 4.484 4.320 4.236 4.072 FP 0.990 = P = 948.2 39.36 14.62 9.074 6.853 5.695 4.995 4.529 4.197 3.950 956.7 39.37 14.54 8.980 6.757 5.600 4.899 4.433 4.102 3.855 963.3 39.39 14.47 8.905 6.681 5.523 4.823 4.357 4.026 3.779 10 968.6 39.40 14.42 8.844 6.619 5.461 4.761 4.295 3.964 3.717 6022 99.39 27.35 14.66 10.16 7.976 6.719 5.911 5.351 4.942 10 6056 99.40 27.23 14.55 10.05 7.874 6.620 5.814 5.257 4.849 f (F ; f1 , f2 ) dF f2 10 4052 98.50 34.12 21.20 16.26 13.75 12.25 11.26 10.56 10.04 5000 99.00 30.82 18.00 13.27 10.92 9.547 8.649 8.022 7.559 5403 99.17 29.46 16.69 12.06 9.780 8.451 7.591 6.992 6.552 5625 99.25 28.71 15.98 11.39 9.148 7.847 7.006 6.422 5.994 f1 5764 5859 99.30 99.33 28.24 27.91 15.52 15.21 10.97 10.67 8.746 8.466 7.460 7.191 6.632 6.371 6.057 5.802 5.636 5.386 5928 99.36 27.67 14.98 10.46 8.260 6.993 6.178 5.613 5.200 5981 99.37 27.49 14.80 10.29 8.102 6.840 6.029 5.467 5.057 514 I Statistical Tables Table I.9: Quantiles tP of Student’s distribution P= tP −∞ f 0.9000 0.9500 0.9750 3.078 6.314 12.706 1.886 2.920 4.303 1.638 2.353 3.182 1.533 2.132 2.776 1.476 2.015 2.571 1.440 1.943 2.447 1.415 1.895 2.365 1.397 1.860 2.306 1.383 1.833 2.262 10 1.372 1.812 2.228 11 1.363 1.796 2.201 12 1.356 1.782 2.179 13 1.350 1.771 2.160 14 1.345 1.761 2.145 15 1.341 1.753 2.131 16 1.337 1.746 2.120 17 1.333 1.740 2.110 18 1.330 1.734 2.101 19 1.328 1.729 2.093 20 1.325 1.725 2.086 30 1.310 1.697 2.042 40 1.303 1.684 2.021 50 1.299 1.676 2.009 60 1.296 1.671 2.000 70 1.294 1.667 1.994 80 1.292 1.664 1.990 90 1.291 1.662 1.987 100 1.290 1.660 1.984 200 1.286 1.653 1.972 500 1.283 1.648 1.965 1000 1.282 1.646 1.962 f (t; f ) dt P 0.9900 0.9950 0.9990 0.9995 31.821 63.657 318.309 636.619 6.965 9.925 22.327 31.599 4.541 5.841 10.215 12.924 3.747 4.604 7.173 8.610 3.365 4.032 5.893 6.869 3.143 3.707 5.208 5.959 2.998 3.499 4.785 5.408 2.896 3.355 4.501 5.041 2.821 3.250 4.297 4.781 2.764 3.169 4.144 4.587 2.718 3.106 4.025 4.437 2.681 3.055 3.930 4.318 2.650 3.012 3.852 4.221 2.624 2.977 3.787 4.140 2.602 2.947 3.733 4.073 2.583 2.921 3.686 4.015 2.567 2.898 3.646 3.965 2.552 2.878 3.610 3.922 2.539 2.861 3.579 3.883 2.528 2.845 3.552 3.850 2.457 2.750 3.385 3.646 2.423 2.704 3.307 3.551 2.403 2.678 3.261 3.496 2.390 2.660 3.232 3.460 2.381 2.648 3.211 3.435 2.374 2.639 3.195 3.416 2.368 2.632 3.183 3.402 2.364 2.626 3.174 3.390 2.345 2.601 3.131 3.340 2.334 2.586 3.107 3.310 2.330 2.581 3.098 3.300 List of Computer Programs∗ AnalysisOfVariance, 318, 319 AuxDer, 426, 428 AuxDri, 230, 426, 428 AuxGrad, 426, 428 AuxHesse, 426, 428 AuxJInputGroup, 427, 429 AuxJNumberInput, 428, 429 AuxJRButtonGroup, 427, 429 AuxZero, 427, 428 DatanFrame, 427, 428 DatanGraphics, 118, 431, 432, 435, 436 DatanMatrix, 348, 399 DatanRandom, 67 DatanUserFunction, 230, 257, 275 DatanVector, 348, 399 E10Mtx, 403 E1Anova, 319 E1Distrib, 106 E1Gr, 443 E1Lsq, 263 E1MaxLike, 173, 427 E1Min, 305 E1Mtx, 399 E1Random, 67 E1Reg, 329 E1Sample, 150 E1Test, 205 E1TimSer, 340 E2Anova, 319 ∗ The E2Distrib, 107 E2Gr, 443 E2Lsq, 263 E2MaxLike, 173 E2Min, 305 E2Mtx, 400 E2Random, 67 E2Reg, 329 E2Sample, 150, 443 E2Test, 205 E2TimSer, 340 E3Distrib, 107 E3Gr, 444 E3Lsq, 263 E3Min, 306 E3Mtx, 400 E3Random, 68 E3Reg, 329 E3Sample, 150, 443 E3Test, 206 E4Gr, 444 E4Lsq, 264 E4Min, 306 E4Mtx, 400 E4Random, 68 E4Reg, 329 E4Sample, 150 E5Gr, 444 E5Lsq, 264 E5Mtx, 401 slanted numbers refer to the Appendix S Brandt, Data Analysis: Statistical and Computational Methods for Scientists and Engineers, DOI 10.1007/978-3-319-03762-2, © Springer International Publishing Switzerland 2014 515 516 E5Sample, 150 E6Gr, 443, 445 E6Lsq, 264 E6Mtx, 401 E6Sample, 151 E7Gr, 443, 445 E7Lsq, 264 E7Mtx, 401 E7Sample, 151 E8Gr, 443, 446 E8Lsq, 265 E8Mtx, 402 E9Lsq, 265 E9Mtx, 402 FunctionOnLine, 275, 303 FunctionsDemo, 414, 422 Gamma, 422, 441 GraphicsWith2DScatterDiagram, 120, 150, 443 GraphicsWithDataPointsAndPolyline, 443 GraphicsWithDataPointsAnd− MultiplePolylines, 443 GraphicsWithHistogramAndPolyline, 443 GraphicsWithHistogram, 118, 150, 443 Histogram, 149, 150 LsqAsg, 263, 265 LsqAsm, 243, 262 LsqAsn, 243, 262, 264 LsqGEn, 264 LsqGen, 255, 257, 262, 264, 265 LsqLin, 224, 261, 263 LsqMar, 235, 243, 262, 264 LsqNon, 230, 235, 243, 261, 263 List of Computer Programs LsqPol, 223, 261, 263 MinAsy, 298, 305, 306 MinCjg, 291, 304, 305 MinCombined, 280, 303 MinCov, 297, 305 MinDir, 280, 281, 303 MinEnclose, 280, 303 MinMar, 294, 304, 305 MinParab, 303 MinPow, 288, 304, 305 MinQdr, 292, 304, 305 MinSim, 283, 303, 305 Regression, 329 S1Anova, 484 S1Distrib, 472 S1Lsq, 475 S1MaxLike, 473 S1Min, 483 S1Random, 470 S1Reg, 485 S1TimSer, 485 S2Anova, 484 S2Lsq, 476 S2MaxLike, 473 S2Min, 483 S2Random, 470 S2Reg, 485 S2TimSer, 486 S3Lsq, 478 S3Random, 471 S4Lsq, 479 S5Lsq, 479 S6Lsq, 481, 482 Sample, 115, 149, 150 SmallSample, 149, 150 StatFunct, 414, 427 TimeSeries, 340 Index∗ a posteriori probability, 153 acceptance probability, 188 addition of errors quadratic, 105 alternative hypothesis, 186, 494 analysis of variance, 307, 500 model, 312 one-way, 307 table, 310 two-way, 311 crossed classification, 483 nested classification, 484 and, 10 ANOVA, 307 table, 310 asymmetric error, 167, 298 asymptotically unbiased estimator, 165 average moving, 332, 501 background, 142 backward substitution, 374 basis, 351 beta function, 418 incomplete, 420 bias, 157, 494 bidiagonal matrix, 350 binomial coefficient, 406, 418 binomial distribution, 70, 409, 451 parameter estimation, 163 binomial theorem, 406 bit, 42 bracketing the minimum, 275 ∗ The Breit–Wigner distribution, 25, 57, 470 byte, 43 Cauchy distribution, 23 central limit theorem, 90, 491 characteristic equation, 378 χ -distribution, 130, 412, 453, 510 quantiles, 511 χ -test, 199, 206, 455 Cholesky decomposition, 372, 401 Cholesky inversion, 401 classification, 307, 500 crossed, 313 nested, 313, 316 one-way, 311 two-way, 313 clipping region, 436 cofactor, 361 color index, 436 column space, 352 column vector, 348 combination, 405 combinatorics, 405 computing coordinates, 433 confidence ellipsoid, 100, 240, 297 interval, 326, 336 level, 133, 494 limits, 297, 298 region, 170, 241, 260, 297, 479 conjugate directions, 285 consistent estimator, 111 constrained measurements, 243, 258 constraint, 396, 403 equations, 244 slanted numbers refer to the Appendix S Brandt, Data Analysis: Statistical and Computational Methods for Scientists and Engineers, DOI 10.1007/978-3-319-03762-2, © Springer International Publishing Switzerland 2014 517 518 contingency table, 203, 456 continued fraction, 418 contour line, 438 controlled variable, 218 convolution, 101, 452, 491 of uniform distribution and normal distribution, 472 of uniform distributions, 102, 471 with the normal distribution, 103 coordinate cross in graphics, 440 correlation coefficient, 29, 489 correlation coefficient of a sample, 473 counted number, 137 covariance, 29, 32, 489 ellipse, 97, 306 ellipsoid, 99, 240 matrix, 297, 489 weighted, 393 critical region, 177, 187, 494 decile, 22 degrees of freedom, 128, 238 derivative logarithmic, 156 determinant, 360 device coordinates, 433 diagonal matrix, 349 dice, 11 direct measurements of different accuracy, 210 of equal accuracy, 209 direct sum, 353 dispersion, 19 distribution binomial, 70, 409, 451 Breit–Wigner, 57, 470 Breit–Wigner , 25 Cauchy, 23 χ , 412, 453, 510 quantiles, 511 F , 413 quantiles, 512 frequency, 109 function, 16, 487 of sample, 110 of several variables, 488 of several variables, 30 of two variables, 25 Gaussian, 86 hypergeometric, 74, 409 Index Lorentz , 25 multivariate normal, 452 multivariate normal, 94 normal, 86, 410, 451, 505, 507 quantiles, 508, 509 of a continuous variable, 492 of a discrete variable, 491 Poisson, 78, 410, 451 Polya, 77 standard normal, 84, 410 quantiles, 508, 509 Student’s, 182, 413 quantiles, 514 t, 413 quantiles, 514 triangular, 58, 448, 470 uniform, 22 unimodal, 21 efficient estimator, 111 eigenvalue, 376 equation, 376 eigenvector, 376 elements, 244 equations of constraint, 238 error asymmetric, 167, 173, 242, 260, 298, 306, 479 combination, 164 of mean, 113 of sample variance, 113 of the first kind, 187, 495 of the second kind, 187, 495 standard, 89 statistical, 73, 106, 137 symmetric, 297, 306 error bars, 117 error function, 411 error model of Laplace, 92 error propagation, 37, 450, 490 E space, 186 estimation, 111 estimator, 111, 493, 494 asymptotically unbiased, 165, 494 consistent, 111, 493 efficiency of, 452 efficient, 111 minimum variance, 161 unbiased, 111, 493 unique, 494 event, 8, 487 Index expectation value, 17, 31, 488, 489 experiment, factorial, 418 F -distribution, 413 quantiles, 512 fit of a Breit–Wigner-function, 478 of a polynomial, 473 of a power law, 475 of an arbitrary linear function, 224 of an exponential function, 232 of a Breit–Wigner function, 480 of a circle to points with measurement errors in abscissa and ordinate, 481 of a Gaussian, 231, 263 of a nonlinear function, 228 of a polynomial, 222, 263 of a proportionality, 224, 263 of a straight line, 218 of a straight line to points with measurement errors in abscissa and ordinate, 264 of a sum of exponential functions, 233 of a sum of two Gaussians and a polynomial, 235, 264 fluctuation statistical, 106 forward substitution, 374 frequency, 106, 490 frequency distribution, 109 F -test, 177, 205, 455 full width at half maximum (FWHM), 119 functional equation, 415 FWHM, 119 Galton’s board, 107 gamma function, 415 incomplete, 420 Gauss–Markov theorem, 238 Gaussian distribution, 86 multivariate, 94 Gaussian elimination, 367 with pivoting, 369 Givens transformation, 354 go button, 427 golden section, 278 519 graphics, 431 class, 431 workstation, 431 Hessian matrix, 271 histogram, 117, 454 bin width, 117 determination of parameters from, 306 Householder transformation, 356, 400 hypergeometric distribution, 74, 409 hyperplane, 353 hypothesis, 175, 186 alternative, 186, 494 composite, 186, 494 null, 186, 494 simple, 186, 494 test of, 494 implementation, 50 incomplete beta function, 420 incomplete gamma function, 420 independence of events, 11 of random variables, 26, 31 indirect measurements linear case, 214 nonlinear case, 226 information, 160, 454 inequality, 157, 161, 494 of a sample, 494 input group, 427 interaction, 315 inverse matrix, 365 Jacobian determinant, 35, 490 of an orthogonal transformation, 40 kernel, 352 Kronecker symbol, 128 Lagrange function, 248 Lagrange multipliers, 126, 247 Laplace error model, 92 law of large numbers, 73, 490 LCG, 45 520 least squares, 209, 362, 497 according to Marquardt, 394 constrained measurements, 243, 498 direct measurements, 209, 499 general case, 251, 498 indirect measurements, 214, 226, 499 properties of the solution, 236 with constraints, 396, 403 with change of scale, 393 with weights, 392 likelihood equation, 155, 494 function, 154, 493 logarithmic, 155, 493 ratio, 154 likelihood-ratio test, 194, 495 linear combination, 351 linear system of equations, 362 Lorentz distribution, 25 lotto, 12 LR-decomposition, 369 main diagonal, 349 mantissa, 43 mapping, 352 marginal probability density, 26, 31, 488 Marquardt minimization, 292 matrix, 348 addition, 348 adjoint, 361, 366 antisymmetric, 350 bidiagonal, 350 diagonal, 349 equations, 362 inverse, 365 main diagonal, 349 multiplication by a constant, 348 by a matrix, 348 norm, 350 null, 349 orthogonal, 354 product, 348 pseudo-inverse, 375 rank, 352 singular, 352, 361 subtraction, 348 symmetric, 350 transposition, 348 Index triangular, 350 tridiagonal, 350 unit, 349 maximum likelihood, 493 maximum-likelihood estimates, 454 mean, 17 error, 113 of sample, 111 mean square, 128 mean square deviation, 128 median, 21, 488 minimization, 267 combined method by BRENT, 280 along a direction, 280 along chosen directions, 287 along the coordinate directions, 284 bracketing the minimum, 275 by the method of POWELL, 287 choice of method, 295 errors in, 296 examples, 298 golden section, 277 in the direction of steepest descent, 271, 288 Marquardt procedure, 292 quadratic interpolation, 280 simplex method, 281 with the quadratic form, 292 minimum variance bound, 161 estimator, 161 MLCG, 45 mode, 21, 488 moments, 18, 27, 31, 487, 489 about the mean, 18 Monte Carlo method, 41 for simulation, 66 minimization, 483 of integration, 64 moving average, 332, 501 Neyman–Pearson lemma, 191, 495 norm Euclidian, 350 normal distribution, 86, 410, 451, 505, 507 multivariate, 63, 94, 452 quantiles, 508, 509 standard, 84, 410 normal equations, 365 null hypothesis, 186, 494 Index 521 null matrix, 349 null space, 352 null vector, 349 number-input region, 428 quadratic average of individual errors, 164 quantile, 22, 488 quartile, 22 one-sided test, 176 operating characteristic function, 188, 495 orthogonal, 349 orthogonal complements, 353 orthogonal matrix, 354 orthogonal polynomials, 500 orthogonal transformation, 354 radio-button group, 427 random component, 332 random number generator, 44 linear congruential (LCG), 45 multiplicative linear congruential (MLCG), 45 random numbers, 41 arbitrarily distributed, 55 acceptance-rejection technique, 59 transformation procedures, 56 normally distributed, 62 random variable, 15, 487 continuous, 15 discrete, 15 rank, 352 ratio of small numbers of events, 144 with background, 147 reduced variable, 20, 488 regression, 321 curve, 325 polynomial, 325, 500 representation of numbers in a computer, 42 row space, 352 row vector, 348 parabola through three points, 273 Pascal’s triangle, 92, 406 permutation, 405 permutation transformation, 359 pivot, 369 Poisson distribution, 78, 410, 451 parameter estimation, 162 quantiles, 503 Polya distribution, 77 polyline, 437 polymarker, 437 polynomial orthogonal, 500 polynomial regression, 500 population, 109, 492 portability, 50 power function, 188, 495 precision absolute, 44 relative, 44 primitive element, 46 principal axes, 378 principal axis transformation, 377 probability, 487 a posteriori, 153 conditional, 10 density, 16, 487 conditional, 26 joint, 488 marginal, 26, 31, 488 of several variables, 31 of two variables, 26 frequency definition, total, 10 pseudo-inverse matrix, 375 sample, 74, 109, 492 correlation coefficient, 473 distribution function, 110 error of variance, 113 of mean, 113 from a bivariate normal distribution, 173 from a continuous population, 111 from finite population, 127 from Gaussian distribution, 130 from partitioned population, 453 from subpopulation, 122 graphical representation, 115 information, 160 mean, 111, 150, 453 random, 110 522 size, 109 small, 136 with background, 142 space, variance, 112, 150, 453 scalar, 349 scalar product, 349 scale factor, 213 scale in graphics, 439 scatter diagram, 150 one-dimensional, 116 two-dimensional, 120 seed, 52 sign inversion, 359 signal, 142 significance level, 175, 494 simplex, 282 singular matrix, 352, 361 singular value, 379 singular value analysis, 380, 383 singular value decomposition, 379, 385, 401, 402 skewness, 20, 488 small numbers of events, 136 ratio, 144 with background, 147 small sample, 136 span, 352 standard deviation, 19, 89, 488 standard normal distribution, 410 statistic, 111, 493 test, 187 statistical error, 73, 137 statistical test, 175 steepest descent, 288 step diagram, 117 Student’s difference test, 184 Student’s distribution, 182, 413 quantiles, 514 Student’s test, 180, 205, 455 subpopulation, 122 subspace, 352 sum of squares, 127, 308 symmetric errors, 297 system of equations linear, 362 triangular, 368 underdetermined, 362 t-distribution, 182, 413 quantiles, 514 Index test, 175, 494, 496 χ , 199, 206, 455, 495 likelihood-ratio, 194, 495 most powerful, 188, 495 one-sided, 176 statistic, 176, 187, 495 Student’s , 205 two-sided, 176 unbiased, 188, 495 uniformly most powerful, 188 text in plot, 441 three-door game, 13 time series, 331 analysis, 331, 501 extrapolation, 485 discontinuities, 485 transformation Givens, 354, 400 Householder, 356, 400 linear, 36 of a vector, 352 of variables, 33, 449, 489 orthogonal, 39, 354 permutation, 359 principal axis, 377 sign inversion, 359 transposition, 348 trend, 332, 501 triangular distribution, 58, 448, 470 triangular matrix, 350 triangular system of equations, 368 tridiagonal matrix, 350 two-sided test, 176 × table, 204 × table test, 204 underdetermined system of equations, 362 uniform distribution, 22 unit matrix, 349 unit vector, 350 variance, 27, 32, 488, 489 of sample, 112 of an estimator, 454 of a random variable, 18 vector, 348 absolute value, 350 components, 348 norm, 350 Index null, 349 row, 348 space, 351 basis, 351 closed, 351 dimension, 351 transformation, 352 unit, 350 vectors linearly dependent, 351 523 linearly independent, 351 orthonormal, 351 viewport, 433 weight matrix, 215 weighted covariance matrix, 393 width full at half maximum, 119 Wilks theorem, 195, 495 window, 433 world coordinates, 432 ... D formulas and methods for gamma and related functions are given Appendix E describes further methods for numerical differentiation, for the determination of zeros, and for interactive input and. .. for the event x < x S Brandt, Data Analysis: Statistical and Computational Methods for Scientists and Engineers, DOI 10.1007/978-3-319-03762-2 3, © Springer International Publishing Switzerland...Data Analysis Siegmund Brandt Data Analysis Statistical and Computational Methods for Scientists and Engineers Fourth Edition Translated by Glen Cowan 123 Siegmund Brandt Department of Physics